In Fig. REF  we show the first and the second fluctuations in momentum \(\mathbf {k}\)  as functions of the spacial dimension \(d\)  (left) and the relative momentum fluctuation \(\Upsilon _k\) , both in the FFG and the HMT models.
One can see from the figure: (a) as the dimension \(d\)  decreases both the first and the second fluctuations decrease, while the relative fluctuation \(\Upsilon _k\)  becomes larger, i.e., low-dimensional nucleonic matter has larger (relative) momentum fluctuation compared with the high dimensional situations, for either the FFG or the HMT model; (b) when the SRC-induced HMT is considered, the fluctuation at a fixed \(d\)  increases, i.e., the interactions strengthen the momentum fluctuations as now nucleons are allowed to move into the high-momentum states above the Fermi surface.
Obviously the moment \(\langle k^{\gamma }\rangle ^{1/\gamma }\)  is limited by the Fermi momentum \(k_{\mathrm {F}}\)  in the FFG model since no nucleons are allowed to be above the Fermi surface. The finding that fluctuations due to interactions are stronger in low dimensions is well known in solid state physics, as the Mermin-Wagner theorem [1]} states that continuous symmetries cannot be spontaneously broken at finite temperatures in systems with short-range interactions in dimensions \(d\le 2\) .
More intuitively, this means that long-range fluctuations can be created with small energy costs.
In addition, we also find that the relative fluctuation \(\Upsilon _k\)  in low dimensions approaches its FFG counterpart (although both are large quantitatively), i.e., \(\Upsilon _k\)  approaches the free model prediction.
More quantitatively, we have \(\Upsilon _k^{\mathrm {HMT}}(4)\approx 0.406\) , \(\Gamma _k^{\mathrm {HMT}}(3)\approx 0.442\) , \(\Gamma _k^{\mathrm {HMT}}(2)\approx 0.467\)  and \(\Gamma _k^{\mathrm {HMT}}(1)\approx 0.632\) . Compared with \(\Upsilon _k^{\mathrm {FFG}}(4)\approx 0.204\) , \(\Upsilon _k^{\mathrm {FFG}}(3)\approx 0.258\) , \(\Upsilon _k^{\mathrm {FFG}}(2)\approx 0.354\)  and \(\Gamma _k^{\mathrm {FFG}}(1)\approx 0.577\) , the HMT enhancements are about 99%, 71%, 32% and 9.5%, respectively as \(d\)  decreases from 4 to 1.
